Question

if y= log [sqrt 1-cos(3x/2)/1=cos(3x/2)] find dy/dx

Answer 1

Step-by-step explanation:

We have,,

$y = log( \sqrt{\frac{1 - \cos( \frac{3x}{2} ) }{1 + \cos( \frac{3x}{2} ) }} )$

$y = log( \sqrt{\frac{2 \sin^{2} ( \frac{3x}{4} ) }{2 \cos ^{2} ( \frac{3x}{4} ) }} )$

$\implies \: y = log( \frac{ \sin( \frac{3x}{4} ) }{ \cos ( \frac{3x}{4} ) } )$

$\implies \: y = log( \tan( \frac{3x}{4} ) )$

$\implies \frac{dy}{dx} = \frac{3}{4} . \frac{1}{ \tan( \frac{3x}{4} ) } . \sec^{2} ( \frac{3x}{4} )$

$\implies \frac{dy}{dx} = \frac{3}{4} . \cot( \frac{3x}{4} ) . \sec^{2} ( \frac{3x}{4} )$

$\implies \frac{dy}{dx} = \frac{3}{4 \sin( \frac{3x}{4} ) \cos( \frac{3x}{4} ) }$

$\implies \frac{dy}{dx} = \frac{3}{2 \sin( \frac{3x}{2} ) }$

$\implies \frac{dy}{dx} = \frac{3}{2 }\cosec( \frac{3x}{2} )$